Research Article Selective Trunk with Multiserver Reservation

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1 Advances in Opeations Reseach Volume 2016, Aticle ID , 8 pages Reseach Aticle Selective Tun with Multiseve Resevation Bacel Maddah 1 and Muhammad El-Taha 2 1 Depatment of Industial Engineeing and Management, Faculty of Engineeing and Achitectue, Ameican Univesity of Beiut, Beiut , Lebanon 2 Depatment of Mathematics and Statistics, Univesity of Southen Maine, 96 Falmouth Steet, Potland, ME , USA Coespondence should be addessed to Bacel Maddah; bacel.maddah@aub.edu.lb Received 3 Novembe 2015; Revised 28 Januay 2016; Accepted 29 Mach 2016 Academic Edito: Yiqiang Zhao Copyight 2016 B. Maddah and M. El-Taha. This is an open access aticle distibuted unde the Ceative Commons Attibution License, which pemits unesticted use, distibution, and epoduction in any medium, povided the oiginal wo is popely cited. We conside a queueing model that is pimaily applicable to taffic contol in communication netwos that use the Selective Tun Resevation technique. Specifically, conside two taffic steams competing fo sevice at an n-seve queueing system. Jobs fom the potected steam, steam 1, ae bloced only if all n seves ae busy. Jobs fom the best effot steam,steam2,aeblocedif n, 1, seves ae busy. Bloced jobs ae diveted to a seconday goup of c nseves with, possibly, a diffeent sevice ate. We extend the liteatue that studied this system fo the special case of =1and pesent an explicit computational scheme to calculate the joint pobabilities of the numbe of pimay and seconday busy seves and elated pefomance measues. We also ague that the model can be useful fo bed allocation in a hospital. 1. Intoduction In this pape, we conside a simple and effective taffic contol technique fo communication netwos with two pioity taffic classes. The method, efeed to as Selective Tun Resevation(STR),assumestwotafficsteamscompetingfo sevice at an n-seve queueing system. Jobs fom steam 1, the potected steam, ae bloced if all n seves ae busy. Jobs fomsteam2, the best effot steam, ae bloced if n, 1, seves ae busy. Bloced jobs ae sent to a seconday goup of c nseves with, possibly, a diffeent sevice ate fom that of the pimay seves. Jobs fom both steams ae lost when all c seves ae busy. El-TahaandHeath[1]consideasimilapoblembutthey estict attention to the case =1and descibe a pocedue to evaluate joint pobabilities fo >1. Thei pape includes significant enhancements of ealie wo (e.g., [1 7]). This pape is an extension of El-Taha and Heath [1] wo and a esponse to inquiies about implementing the STR method of [1] fo all 1. In paticula, we explicitly deive the joint pobabilities fo all 1and povide an efficient algoithm fo computing them. The bloc diagam of the model in this pape is shown in Figue 1. We deive the joint pobabilities of the numbe of pimay and seconday busy seves. Fom these pobabilities, pefomance measues, including oveflow pobabilities of the seve goups, ae deived. We then apply ou esults to the specific case of two Poisson taffic steams and two heteogeneous goups of exponential seves. One altenative application of the model in this pape, which came to ou attention ecently, is fo bed allocation in a hospital wad. The liteatue contains seveal examples of bed allocation with inteesting similaities to the classic STR poblem of communication. Examples of these wos include [8 13]. In many of these wos (e.g., [8, 10]), two types of patients ae consideed, seious and nonseious, which aive accoding to a Poisson pocess and have an exponentially distibuted duation of stay (sevice time), simila to the two steams of aivals in the STR poblem. Moe inteestingly, this liteatue discusses the setting of a cut-off occupancy in tems of a numbe of beds in the wad eseved fo seious patients only, which is in the same spiit of STR. Esogbue and Singh [10] futhe conside buffe accommodation which mimics the seconday seves we

2 2 Advances in Opeations Reseach 1 λ 2. λ 1 n. n Pimay seves Oveflow (steam 2) 1 Oveflow (steam 1). Loss (steam 2) Loss (steam 1) c n Seconday seves Figue 1: Bloc diagam of the STR model. conside in this pape. Howeve, one distinctive diffeence between STR in communication and bed allocation in hospitals is that, in the latte, the sevice times depend on the type of aival. This equie state expansion, to tac patient type, in Maovian STR models, such as the one in this pape. We believe that ou efficient computational schemes can be tailoed fo such expanded-state bed allocation poblems. The pape is oganized as follows. In Section 2, we pesent an efficient iteative method to find the joint pobability distibution of the numbe of busy pimay and seconday seves. In Section 3, we povide an application that consides two Poisson steams, based on the analysis of Section 2, and we illustate the application of the method by a numeical example. In Section 4, we daw some conclusions and discuss possible extensions. Finally, in the Appendix we povide an algoithm that is used fo the development of a compute pogam of the pesent model. 2. Joint Stationay Pobabilities Fomally, we descibe the above model as a two-dimensional stochastic pocess X {(X 1 (t), X 2 (t)); t 0 with finite state space S such that X 1 (t) taes intege values in {0,...,n, n 1; X 2 (t) taes intege values in {0,...,c n, c n+1. Hee, X 1 (t) epesents the numbe of pimay busy seves and X 2 (t) epesents the numbe of seconday busy seves. Utilizing a simila notation to El-Taha and Heath [1], define β 1 (i, j) to be the state (i, j) steam 1 aival ate and β 2 (i, j) tobethesteam2aivalatewhenthesystemstate is (i, j). Letβ(i, j) = β 1 (i, j) + β 2 (i, j). Defineγ 1 (i) to be the pimay goup sevice ate when i pimay seves ae busy and γ 2 (j) to be the seconday goup sevice ate when j seconday seves ae busy. Let p i,j denote the state (i, j) joint stationay pobability, i = 0,...,n, j = 0,...,c n. The tansition ate out of state (i, j) is denoted by α(i, j) = β(i, j)+γ 1 (i)+γ 2 (j).then,thestatepobabilitiesfoany 1 satisfy the global balance equations: α (i, j) p i,j =β(i 1,j)p i 1,j +γ 1 (i+1) p i+1,j +γ 2 (j + 1) p i,j+1, i=0,1,...,n 1; j=c n,...,0, α(n,j)p n,j =β(n 1,j)p n 1,j +β 2 (n, j 1) p n,j 1 +γ 1 (n +1) p n +1,j +γ 2 (j + 1) p n,j+1 ; j=c n,...,0, (1) (2)

3 Advances in Opeations Reseach 3 α (i, j) p i,j =β 1 (i 1, j) p i 1,j +β 2 (i, j 1) p i,j 1 +γ 1 (i+1) p i+1,j +γ 2 (j + 1) p i,j+1 ; i=n +1,...,n 1; j=c n,...,0, (3) The nomalization equation is n i=0 c n j=0 p i,j = 1.Theelations in (1) (4) ae gaphically illustated in Figue 2. Equations (1) (4) expand simila flow balance equations in El-Taha and Heath [1] by allowing fo >1in (3). Equations (1), (2), and (4) ae the same as those in [1]. We pesent these equations hee fo completeness. The emainde of this section is devoted to solving (1) (4) efficiently via an iteative scheme. The fist step, in (5), expesses each p i,j, i=1,...,n, j=0,...,c n,intems of the pobabilities p 0,, =j,...,c n: α(n,j)p n,j =β 1 (n 1, j) p n 1,j +β(n,j 1)p n,j 1 +γ 2 (j + 1) p n,j+1 ; (4) p i,j = min{i,c n j ( 1) G (i, j, ) p 0,j+, i=1,...,n, j=0,...,c n, (5) j=c n,...,0. whee, fo each j=c n,...,0, G (i, j, ) 1 i=0, ; { = γ 1 (i) 1 [α (i 1, j) G (i 1, j, ) β (i 2, j) G (i 2, j, ) + γ 2 (j+1)g(i 1,j+1, 1)],,...,min {i,c n j, i=1,...,n ; { { 0 othewise. (6) Equation (5) can be poved in a simila manne to Lemma 2.1 of El-Taha and Heath [1]. In ou main esult, Theoem 1, (5) is utilized in obtaining {p i,j, i = 0,...,n ; j = 0,...,c n, in tems of p n,c n, = 0,1,...,. The p n,c n pobabilities ae obtained by solving a system of +1linea equations fomulated by applying Theoem 1 in combination with the limiting state pobabilities of the numbe of busy pimay seves. Finally, all joint pobabilities ae computed by appealing again to Theoem 1 and by eplacing p n,c n,,1,...,,bytheicomputedvalues. Theoem 1 is ou main contibution with espect to El- Taha and Heath [1], as it gives explicit expessions of the limiting pobabilities p i,j intemsofasmallesetofthese pobabilities, p n,c n, = 0,1,...,. The main esult in [1] is a special case of Theoem 1 with =1. Theoem 1. Fo all (i,j), i=0,...,nand j=0,...,c n, whee p i,j = min(i,c n j) B m (i, j) = B m (i, j) p n +m,c n, (7) ( 1) G (i, j, ) B m (0, j + ),,...,; i=1,...,n 1; n 2; j=c n 1,...,0; and the bounday conditions ae B m (0, j) = G (n, j, 0) 1 [B m (n, j) (8) min(n,c n j) =1 B m (n, j) = β 2 (n, j) 1 ( 1) G(n,j,)B m (0, j + )], [α(n,j+1)b m (n, j + 1),...,; j=c n 1,...,0, β(n 1,j+1)B m (n 1,j+1) γ 1 (n +1) B m (n +1,j+1) γ 2 (j + 2) B m (n,j+2)],,...,; j=c n 1,...,0, B m (i, j) = β 2 (i, j) 1 [α (i, j + 1) B m (i, j + 1) β 1 (i 1, j + 1) B m (i 1,j+1) γ 1 (i+1) B m (i + 1, j + 1) γ 2 (j + 2) B m (i, j + 2)],,...,; i=n +1,...,n 1; j=c n 1,...,0, B m (n, j) = β (n, j) 1 [α (n, j + 1) B m (n, j + 1) β 1 (n 1, j + 1) B m (n 1,j+1) γ 2 (j + 2) B m (n, j + 2)],,...,; j=c n 1,...,0, (9) (10) (11) (12)

4 4 Advances in Opeations Reseach γ 2 (1) γ 2 (2) γ 2 (c n) 0, 0 0, 1 0, 2 0, j 0, c n 1 0, c n i 1, j β(i 1, j) γ 2 (j) γ 1 (i) γ 2 (j + 1) i, 0 i, j 1 i, j i, j + 1 i, c n β(i, j) γ 1 (i + 1) i + 1, j n 1, j β(n 1, j) γ 1 (n ) n, 0 γ 2 (j) γ 2 (j + 1) n,j 1 n,j n,j+1 n, c n β 2 (n, j 1) β 1 (n, j) β 2 (n, j) γ 1 (n + 1) n + 1, j i 1,j β 1 (i 1,j) γ 1 (i ) γ 2 (j) γ 2 (j + 1) i,0 i,j 1 i,j i,j+1 i,c n β 2 (i,j 1) β 1 (i,j) β 2 (i,j) γ 1 (i +1) i +1,j n 1, j β 1 (n 1, j) γ 1 (n) n, 0 γ 2 (j) γ 2 (j + 1) n, j 1 n, j n, j + 1 β(n, j 1) β(n, j) Figue 2: Tansition diagam of the model. n, c n

5 Advances in Opeations Reseach 5 B m (i, c n) G (i, c n, 0) { G (n,c n,0) ; ; i=0,...,n = 1; i=n +m; m=1,...,; { { 0; othewise. (13) Poof. The joint pobabilities ae deived iteatively. Fist conside j=c n.by(5), p i,c n =G(i, c n, 0) p 0,c n ; i=0,1,...,n, (14) which implies p n,c n =G(n,c n,0) p 0,c n, (15) and then p 0,c n =G(n,c n,0) 1 p n,c n. (16) It follows that G (i, c n, 0) p i,c n = G (n,c n,0) p n,c n (17) =B 0 (i, c n) p n,c n, i=0,...,n, whee B 0 (i, c n) is given by (13). Note that, fo i=n + 1,...,nand all m = 0,...,, B m (i, c n) = 0 except B m (n +m,c n)=1. Theefoe, the theoem is valid fo j=c n. Next, we conside the geneal case. Rewite (3) as p i,j 1 =β 2 (i, j 1) 1 [α (i, j) p i,j β 1 (i 1, j) p i 1,j γ 1 (i+1) p i+1,j γ 2 (j + 1) p i,j+1 ]. Using a change of vaiable (j 1 j), we obtain p i,j =β 2 (i, j) 1 [α (i, j + 1) p i,j+1 β 1 (i 1, j + 1) p i 1,j+1 γ 1 (i+1) p i+1,j+1 (18) (19) γ 1 (i+1) B m (i + 1, j + 1) γ 2 (j + 2) B m (i, j + 2)] p n +m,c n = B m (i, j) p n +m,c n, (21) whee B m (i, j) is given by (11). Similaly, by manipulating (2) and (4) one can show that p n,j = B m(n, j)p n +m,c n, j = 0,...,c n, wheeb m (n, j) is given by (10) and that p n,j = B m(n, j)p n +m,c n, j = 0,...,c n,wheeb m (n, j) is given by (12). Next, we evaluate p 0,j, j=c n 1,...,0.Itfollowsfom the above equations and (5) that p n,j = = B m (n, j) p n +m,c n min(n,c n j) ( 1) G(n,j,)p 0,j+ =G(n,j,0)p 0,j min(n,c n j) + =1 ( 1) G(n,j,)p 0,j+. (22) Solving fo p 0,j, using the induction agument, and simplifying, p 0,j =G(n,j,0) 1 min(n,c n j) [p n,j =1 G(n,j,)p 0,j+ ]=G(n,j,0) 1 ( 1) γ 2 (j + 2) p i,j+2 ], whee i=n +1,...,n 1;andj=c n 1,...,0. Now, suppose that the theoem holds fo j+1,j+2,...,c n.thenfoi=n +1,...,n 1;andj=c n 1,...,0 p i,j =β 2 (i, j) 1 {α (i, j + 1) B m (i, j + 1) p n +m,c n β 1 (i 1,j+1) B m (i 1,j+1)p n +m,c n γ 1 (i+1) B m (i+1,j+1)p n +m,c n γ 2 (j + 2) B m (i, j + 2) p n +m,c n, which leads to p i,j =β 2 (i, j) 1 { [α (i, j + 1) B m (i, j + 1) β 1 (i 1,j+1)B m (i 1, j + 1) (20) [ B m (n, j) p n +m,c n min(n,c n j) ( 1) G(n,j,)B m (0, j + ) =1 p n +m,c n ]=G(n,j,0) 1 min(n,c n j) =1 p n +m,c n = [B m (n, j) ( 1) G(n,j,)B m (0, j + )] B m (0, j) p n +m,c n, whee B m (0, j) is given by (9). Now, fo i=1,2,...,n 1, using (5), p i,j = = min(i,c n j) ( 1) G (i, j, ) p 0,j+ min(i,c n j) ( 1) G (i, j, ) (23)

6 6 Advances in Opeations Reseach B m (0, j + ) p n +m,c n = [ min(i,c n j) p n +m,c n = ( 1) G (i, j, ) B m (0, j + )] B m (i, j) p n +m,c n, (24) whee B m (i, j) is given by (8). The induction agument completes the poof of the theoem. Now, we poceed to evaluate p n +m,c n, m = 0,...,. In the following theoem, these ae obtained based on the maginal limiting pobabilities of the numbe of busy pimay seves, p (n), = 0,1,...,n, which have a simple closed fom. Theoem 2. The pobabilities p n +m,c n,,...,,aethe solution to the following system of linea equations: C (m, ) p n +m,c n =p (n), (25) whee p (n) = ( i=1 β(i 1)γ 1(i) 1 )/(1 + n l=1 l i=1 β(i 1)γ 1 (i) 1 ), C(m, ) = c n j=0 B m(, j), and B m (, j), = 0,1,...,n, j=0,1,...,c nae as given in Theoem 1. Poof. The poof follows by noting that p (n) 0,1,...,n. In addition, the pobabilities p (n) balance equations: β () p (n) = c n j=0 p,j, = satisfy the flow =γ 1 (+1) p (n) +1,,1,...,n. (26) Utilizing p (n) 0 as a nomalizing constant and utilizing the expession in Theoem 1 of p,j,,1,...,n, j=0,...,c n,completethepoof. 3. Application Simila to El-Taha and Heath [1], we pesent an application fo the special case of two, state-independent, Poisson aival steams, a eal time (potected)steamwithaivalateλ 1 and a best effot steam with aival ate λ 2.Bothpimayand seconday goups of seves have exponentially distibuted sevice times with constant sevice ates μ 1 and μ 2,espectively.Thetansitionatesaethenobtainedfomthegeneal model as β (i, j) = λ 1 +λ 2, i=0,...,n 1, i=n; j=0,...,c n, β 1 (i, j) = λ 1, i=n,...,n 1, j=0,...,c n, β 2 (i, j) = λ 2, i=n,...,n 1,j=0,...,c n 1, (27) γ 1 (i) =iμ 1, i=0,...,n, γ 2 (j) = jμ 2, j=0,...,c n, β () ={ λ 1 +λ 2,,...,n 1, λ 1, =n,...,n 1. The joint pobabilities, p i,j,aeobtainedfomtheoem1andsubsequentesultsofsection2.theoveflow pobabilities,thelosspobabilitiesofthepotectedandthe best effot steams, and the mean numbe of busy pimay and seconday seves ae then obtained fom the joint pobabilities, p i,j. Fo simplicity, one may use the distibution ofthenumbeofbusypimayseves,p (n) i, as defined in Theoem 2, in obtaining some of these measues of pefomance. We have developed a C++ pogam fo obtaining the joint pobabilitiesandothemeasuesofpefomance.asummay of the pogam algoithm is pesented in the Appendix. To solve the system of linea equations in (25), we use the LU decomposition technique with patial pivoting as pesented by Pess et al. [14] (pp ). Pess et al. [14] epot that this method is thee times faste than the classical Gauss- Jodan elimination method. In addition, the method is stable and yield esults with high pecision fo systems of linea equations of dimension in the ode of couple of hundeds. Theefoe, ou computational scheme is expected to be fast and obust. Example. To illustate the application of ou method, conside an example with (c, n, ) = (16, 10, 3) and aival and sevice ates (λ 1,λ 2,μ 1,μ 2 ) = (10.0, 10.0, 3.0, 1.0).Jointpobabilities, geneated by ou C++ pogam, ae pesented as follows: p i,j ; i=0,1,...,10; j=0,..., (28)

7 Advances in Opeations Reseach 7 INITIALIZE: input c, n,, α(i, j), β 1 (i, j), β 2 (i, j), β(i, j), γ 1 (i), γ 2 (j) 0 i n, 0 j c n; STEP 1. CASE OF (j = c n) G(0, c n, 0) = 1; fo (i = 1; i n ; i++) G(i, c n, 0) = γ 1 (i) 1 [α(i 1, c n)g(i 1, c n, 0) β(i 2, c n)g(i 2, c n, 0)]; fo (i = 0; i n ; i++){ B 0 (i, c n) = G(i, c n, 0)/G(n, c n, 0); fo (m = 1; m ; m++){ B m (n +m,c n)=1; STEP 2. CASE OF (j c n 1) fo (j=c n 1, j 0; j ){ G(0, j, 0) = 1; fo (i = 1; i n ; i++){ fo ( = 0; min(i, c n j); ++) G(i, j, ) = γ 1 (i) 1 [α(i 1, j)g(i 1, j, ) β(i 2, j)g(i 2, j, ) + γ 2 (j + 1)G(i 1, j + 1, 1)] fo (m = 0; m ; m++){ B m (n, j) = β(n, j) 1 [α(n, j + 1)B m (n, j + 1) β 1 (n 1, j + 1)B m (n 1,j+1) γ 2 (j + 2)B m (n, j + 2)]; B m (n, j) = β 2 (n, j) 1 [α(n, j + 1)B m (n,j+1) β(n 1, j + 1)B m (n 1,j+1) γ 1 (n +1)B m (n +1,j+1) γ 2 (j + 2)B m (n,j+2)]; fo (i=n +1; i n 1; i++){ B m (i, j) = β 2 (i, j) 1 [α(i, j + 1)B m (i, j + 1) β 1 (i 1, j + 1)B m (i 1, j + 1) γ 1 (i + 1)B m (i + 1, j + 1) γ 2 (j + 2)B m (i, j + 2)]; B m (0, j) = G(n, j, 0) 1 [B m (n, j) min(n,c n j) ( 1) G(n, j, )B m (0, j + )]; =1 fo (i = 1; i n 1; i++) B m (i, j) = min(i,c n j) COMPUTE JOINT PROBABILITIES: input β(), γ 1 ( + 1) < n; fo ( = n ; n; ++) ( 1) G(i, j, )B m (0, j + ) p (n) = i=1 β(i 1)/γ 1(i)[1 + n =1 i=1 β(i 1)/γ 1(i)] 1 ; Compute: C(m, ) = c n j=0 B m(, j) Solve the system of equations C(m, )p n +m,c n =p (n) (Note; the unnowns ae p n +m,c n,,...,, =n,...,n.) fo (j = c n; j 0; j ) fo (i = 0; i n; i++) p i,j = B m(i, j)p n +m,c n END Algoithm 1 Measues of pefomance ae easily obtained fom the above joint pobabilities o the distibution of the numbe of busy pimay seves, p (n) i. These measues of pefomance aecomputedasfollows:potectedsteamoveflowpobability, P 1 =p (n) n = c n j=0 p n,j = 0.010; best effot steam oveflow pobability, P 2 = n i=n p(n) i = n i=n c n j=0 p i,j = 0.324; potected steam loss pobability, Q 1 =p n,c n = 0.003; best effot steam loss pobability, Q 2 = n i=n p i,c n = 0.052; mean numbe of busy pimay seves, L 1 = n i=1 ip(n) i = n i=1 c n j=0 ip i,j = 5.553; and mean numbe of busy seconday

8 8 Advances in Opeations Reseach seves, L 2 = n i=0 c n j=1 jp i,j = These esults have been validated via discete-event simulation. Note finally that the example hee is selected fo illustative puposes only. Ou computational scheme can be futhe utilized in detemining an optimal value that meets a specific citeion such as a specified oveflow pobability fo potected taffic and studying the effect of inceasing the best effot taffic on the potected taffic quality of sevice. Fo details, see Examples 2 and 3 in El-Taha and Heath [1]. 4. Conclusion We have extended the esults of El-Taha and Heath [1] to explicitly allow multiple eseved channels, >1. As in[1], we pesent an efficient iteative technique to evaluate the joint pobabilities of busy pimay and seconday seves. A new component of the pesent model is the system of +1linea equations given in (25). By using effective numeical methods andwiththeavailabilityofcomputeswithhighpocessing speed and stoage capacity, this pat of the computational scheme will not be of concen even fo lage values of. We have illustated the application of the method fo the case of two Poisson aival steams competing fo sevice on two sets of heteogeneous seves with constant sevice ates. It is impotant to note that the pesent model may be applicable fo moe geneal situations such as aivals fom a finite souce and state dependent sevice ates. This may be an aea fo futhe extensions and applications of the esults of this pape. Anothe aea fo futue wo is to tailo the cuent model to the poblem of bed allocation in a hospital wad, with sevice ates depending on the type of aival, which equies state expansion of the Maov chain. [6] E. A. Van Doon, On the oveflow pocess fom a finite Maovian queue, Pefomance Evaluation, vol. 4, no. 4, pp , [7]J.F.E.WallinandB.Sandes, Thecalculationofoveflow moments in loss systems with selective tun esevation, Pefomance Evaluation,vol.15,no.3,pp ,1992. [8] J.P.Young,A queueing theoy appoach to the contol of hospital in-patient census [Ph.D. dissetation], School of Engineeing, The Johns Hopins Univesity, Baltimoe, Md, USA, [9] P. Kolesa, A maovian model fo hospital admission scheduling, Management Science,vol.16, no.6,pp , [10] A. O. Esogbue and A. J. Singh, A stochastic model fo an optimal pioity bed distibution poblem in a hospital wad, Opeations Reseach, vol. 24, no. 5, pp , [11] E. P. C. Kao and G. G. Tung, Bed allocation in a public health cae delivey system, Management Science, vol.27,no.5,pp , [12] L. V. Geen and V. Nguyen, Stategies fo cutting hospital beds: the impact on patient sevice, Health Sevices Reseach,vol.36, no. 2, pp , [13]J.K.CochanandK.T.Roche, Amulti-classqueuingnetwo analysis methodology fo impoving hospital emegency depatment pefomance, Computes & Opeations Reseach, vol.36,no.5,pp ,2009. [14] W. Pess, S. Teuolsy, W. Vetteling, and B. Flanney, Numeical Recipes in C: The At of Scientific Computing, Cambidge Univesity Pess, Cambidge, UK, Appendix Fo a c-seve system with n pimay seves and m=c n seconday seves, Algoithm 1 computes the joint stationay pobabilities p i,j. Competing Inteests The authos declae that they have no competing inteests. Refeences [1] M. El-Taha and J. R. Heath, Taffic oveflow in loss systems with selective tun esevation, Pefomance Evaluation, vol. 41, no. 4, pp , [2] J. M. Ainpelu, The oveload pefomance of engineeed netwos with nonhieachical and hieachical outing, The Bell System Technical Jounal,vol.63,no.7,1984. [3] A. Giad, Blocing pobability of nonintege goups with tun esevation, IEEE Tansactions on Communications,vol. 33, no. 2, pp , [4] F. Le Gall and J. Benussou, Blocing pobabilities fo tun esevation policy, IEEE Tansactions on Communications,vol. 35,no.3,pp ,1987. [5] L. Paatz, Taffic theoy of cicuit esevation, Austalian Telecommunication Reseach,vol.20,no.2,pp.51 60,1986.

9 Advances in Opeations Reseach Advances in Decision Sciences Jounal of Applied Mathematics Algeba Jounal of Pobability and Statistics The Scientific Wold Jounal Intenational Jounal of Diffeential Equations Submit you manuscipts at Intenational Jounal of Advances in Combinatoics Mathematical Physics Jounal of Complex Analysis Intenational Jounal of Mathematics and Mathematical Sciences Mathematical Poblems in Engineeing Jounal of Mathematics Discete Mathematics Jounal of Discete Dynamics in Natue and Society Jounal of Function Spaces Abstact and Applied Analysis Intenational Jounal of Jounal of Stochastic Analysis Optimization